Integrate x²/e^x dx
We can use integration by parts to solve this integral:
Let u = x² and dv/dx = e^(-x)dx. Then du/dx = 2x and v = -e^(-x).
Using the formula for integration by parts:
∫ u dv/dx dx = uv - ∫ v du/dx dx
Substituting in our values:
∫ x²/e^(x) dx = -x²e^(-x) - ∫ -2xe^(-x) dx
We can use integration by parts again to solve the second integral:
Let u = -2x and dv/dx = e^(-x)dx. Then du/dx = -2 and v = -e^(-x).
Using the formula for integration by parts again:
∫ u dv/dx dx = uv - ∫ v du/dx dx
Substituting in our values:
- ∫ -2xe^(-x) dx = 2xe^(-x) - ∫ -2e^(-x) dx
= 2xe^(-x) + 2e^(-x) + C, where C is the constant of integration.
Substituting this result back into our original equation:
∫ x²/e^(x) dx = -x²e^(-x) - (2xe^(-x) + 2e^(-x) + C)
= -(x² + 2x + 2)e^(-x) + C.
Therefore, the solution to the integral is ∫ x²/e^(x) dx = -(x² + 2x + 2)e^(-x) + C.
To integrate x²/e^x dx, we can use the technique of integration by parts. This method involves finding the antiderivative of a product of two functions.
Let's use the following formula for integration by parts:
∫ u * v dx = u * ∫ v dx - ∫ (u' * ∫ v dx) dx
In this case, let's choose:
u = x² (so that du/dx = 2x)
v = e^x (so that dv/dx = e^x)
Now, let's compute u' and ∫ v dx:
u' = du/dx = 2x
∫ v dx = ∫ e^x dx = e^x
Next, plug these values into the integration by parts formula:
∫ x²/e^x dx = x² * ∫ e^x dx - ∫ (2x * e^x) dx
Simplifying:
∫ x²/e^x dx = x² * e^x - 2 * ∫ (x * e^x) dx
Now, let's focus on the remaining integral:
∫ (x * e^x) dx
We can use integration by parts again to evaluate this integral:
Let's choose:
u = x (so that du/dx = 1)
v = e^x (so that dv/dx = e^x)
Now, let's compute u' and ∫ v dx:
u' = du/dx = 1
∫ v dx = ∫ e^x dx = e^x
Plugging these values into the integration by parts formula:
∫ (x * e^x) dx = x * ∫ e^x dx - ∫ (1 * e^x) dx
Simplifying:
∫ (x * e^x) dx = x * e^x - ∫ e^x dx
We have obtained another integral involving e^x. This integral is the same as the one we started with. So we can substitute this result into the original equation:
∫ x²/e^x dx = x² * e^x - 2 * (x * e^x - ∫ e^x dx)
Simplifying further:
∫ x²/e^x dx = x² * e^x - 2 * (x * e^x - e^x)
Combining like terms:
∫ x²/e^x dx = x² * e^x - 2x * e^x + 2 * e^x
Therefore, the integral of x²/e^x dx is equal to x² * e^x - 2x * e^x + 2 * e^x + C, where C is the constant of integration.